Domination by small sets versus density (Q2215632)

Given an infinite cardinal \(\kappa\), the authors prove that a regular space \(X\) must have density not exceeding \(\kappa\) if and only if every subset of \(X\) of power \((2^\kappa )^+\) is \(\kappa\)-dominated, i.e., contained in the closure of a set of cardinality \(\kappa\). This work is devoted to a study of the spaces in which all subsets of cardinality not exceeding \(2^\kappa\) are \(\kappa\)-dominated; this property is called exponential \(\kappa\)-domination. The spaces with exponential \(\omega\)-domination first appeared in the paper [\textit{V. V. Tkachuk}, Quaest. Math. 43, No. 10, 1391--1403 (2020; Zbl 1478.54028)]. It is proved, among other things, that the class of spaces with exponential \(\kappa\)-domination is preserved by continuous images, open subspaces, \(\Sigma _{2^\kappa}\)-products and \(\kappa\)-unions. It is also shown that the cardinal \(\kappa ^+\) is a caliber of any space \(X\) with exponential \(\kappa\)-domination. A Čech-complete space features exponential \(\kappa\)-domination if and only if its density does not exceed \(\kappa\). However, the density of countably compact spaces with exponential \(\omega\)-domination has no upper bound. If a regular space \(X\) has a dense subspace \(Y\) with \(\pi _\chi(Y ) \leq 2^\kappa\), then \(d(X) \leq \kappa\) if and only if \(X\) features exponential \(\kappa\)-domination. It is included an example of a non-separable Tychonoff space of countable pseudocharacter with exponential \(\omega\)-domination. At the end of the paper, the authors provide a list of interesting open questions.